JUST THE MATHS SLIDES NUMBER 13.13 INTEGRATION APPLICATIONS 13 - PDF document

“JUST THE MATHS” SLIDES NUMBER 13.13 INTEGRATION APPLICATIONS 13 (Second moments of a volume (A)) by A.J.Hobson 13.13.1 Introduction 13.13.2 The second moment of a volume of revolution about the y -axis 13.13.3 The second moment of a volume of revolution about the x -axis

UNIT 13.13 - INTEGRATION APPLICATIONS 13 SECOND MOMENTS OF A VOLUME (A) 13.13.1 INTRODUCTION Let R denote a region (with volume V ) in space and suppose that δV is the volume of a small element of this region Then the “second moment” of R about a fixed line, l , is given by R h 2 δV, lim � δV → 0 where h is the perpendicular distance from l of the element with volume, δV . ✡ ✡ ✡ ✡ ✡ ◗◗◗◗◗◗◗◗ ✡ ✡ R ✡ h ✡ ◗ δV ✡ l ❡ ✡ ✡ ✡ 1

EXAMPLE Determine the second moment, about its own axis, of a solid right-circular cylinder with height, h , and radius, a . Solution r h In a thin cylindrical shell with internal radius, r , and thickness, δr , all of the elements of volume have the same perpendicular distance, r , from the axis of moments. Hence the second moment of this shell is r 2 (2 πrhδr ) . The total second moment is therefore given by 0 2 πhr 3 d r = πa 4 h � a r = a r =0 r 2 (2 πrhδr ) = lim . � 2 δr → 0 2

13.13.2 THE SECOND MOMENT OF A VOLUME OF REVOLUTION ABOUT THE Y-AXIS Consider a region in the first quadrant of the xy -plane, bounded by the x -axis, the lines x = a , x = b and the curve whose equation is y = f ( x ) . y ✻ ✲ x O a δx b The volume of revolution of a narrow ‘strip’, of width δx , and height, y , (parallel to the y -axis), is a cylindrical ‘shell’, of internal radius x , height, y , and thickness, δx . Hence, from the example in the previous section, its sec- ond moment about the y -axis is 2 πx 3 yδx. 3

Thus, the total second moment about the y -axis is given by x = b x = a 2 πx 3 yδx lim � δx → 0 � b a 2 πx 3 y d x. = Note: For the volume of revolution, about the x -axis, of a region in the first quadrant, bounded by the y -axis, the lines y = c , y = d and the curve whose equation is x = g ( y ) , we may reverse the roles of x and y so that the second moment about the x -axis is given by � d c 2 πy 3 x d y. 4

y ✻ d δy c ✲ x O EXAMPLE Determine the second moment, about a diameter, of a solid sphere with radius a . Solution We may consider, first, the volume of revolution about the y -axis of the region bounded in the first quadrant by the x -axis, the y -axis and the circle whose equation is x 2 + y 2 = a 2 , then double the result obtained. 5

y ✻ ✡ ✡ a ✡ ✡ ✡ ✲ x O The total second moment is given by 0 2 πx 3 √ � a a 2 − x 2 d x 2 � π a 3 sin 3 θ.a cos θ.a cos θ d θ, = 4 π 2 0 if we substitute x = a sin θ . This simplifies to 4 πa 5 � π sin 3 θ cos 2 θ d θ 2 0 � π cos 2 θ − cos 4 θ � � = 4 π sin θ d θ, 2 0 if we make use of the trigonometric identity sin 2 θ ≡ 1 − cos 2 θ. 6

The total second moment is now given by π  − cos 3 θ + cos 5 θ   2 4 πa 5     3 5  0  = 8 πa 5  1 3 − 1   = 4 πa 5 15 .   5 13.13.3 THE SECOND MOMENT OF A VOLUME OF REVOLUTION ABOUT THE X-AXIS In the introduction to this Unit, a formula was established for the second moment of a solid right-circular cylinder about its own axis. This result may now be used to determine the second moment, about the x -axis, for the volume of revolution about this axis, of a region enclosed in the first quadrant by the x -axis, the lines x = a , x = b and the curve whose equation is y = f ( x ) . 7

y ✻ ✲ x O a δx b The volume of revolution about the x -axis of a narrow strip, of width δx and height y , is a cylindrical ‘disc’ whose second moment about the x -axis is πy 4 δx . 2 Hence, the second moment of the whole region about the x -axis is given by πy 4 πy 4 � b x = b lim 2 δx = d x. � a 2 x = a δx → 0 EXAMPLE Determine the second moment about the x -axis, for the volume of revolution about this axis of the region, bounded in the first quadrant, by the x -axis, the y -axis, the line x = 1 and the line whose equation is y = x + 1 . 8

Solution y ✻ � � � ✲ x O 1 π ( x + 1) 4 � 1 Second moment = d x 0 2 1  π ( x + 1) 4   = 31 π = 10 .     10  0 Note: The second moment of a volume about a certain axis is closely related to its “moment of inertia” about that axis In fact, for a solid with uniform density, ρ , the moment of inertia is ρ times the second moment of volume, since multiplication by ρ , of elements of volume, converts them into elements of mass 9